A Neumann problem with critical exponent in nonconvex domains and Lin-Ni’s conjecture
HTML articles powered by AMS MathViewer
- by Liping Wang, Juncheng Wei and Shusen Yan PDF
- Trans. Amer. Math. Soc. 362 (2010), 4581-4615 Request permission
Abstract:
We consider the following nonlinear Neumann problem: \[ \left \{\begin {array}{lll} -\Delta u + \mu u = u^{\frac {N+2}{N-2}},\quad u>0 \quad & \mbox {in} \ \Omega , \\ \frac {\partial u}{\partial n}=0 & \mbox {on} \ \partial \Omega , \end {array}\right .\] where $\Omega \subset \mathbb {R}^N$ is a smooth and bounded domain, $\mu > 0$ and $n$ denotes the outward unit normal vector of $\partial \Omega$. Lin and Ni (1986) conjectured that for $\mu$ small, all solutions are constants. We show that this conjecture is false for all dimensions in some (partially symmetric) nonconvex domains $\Omega$. Furthermore, we prove that for any fixed $\mu$, there are infinitely many positive solutions, whose energy can be made arbitrarily large. This seems to be a new phenomenon for elliptic problems in bounded domains.References
- Adimurthi and G. Mancini, The Neumann problem for elliptic equations with critical nonlinearity, Nonlinear analysis, Sc. Norm. Super. di Pisa Quaderni, Scuola Norm. Sup., Pisa, 1991, pp. 9–25. MR 1205370
- Adimurthi and G. Mancini, Geometry and topology of the boundary in the critical Neumann problem, J. Reine Angew. Math. 456 (1994), 1–18. MR 1301449, DOI 10.1515/crll.1994.456.1
- Adimurthi, Filomena Pacella, and S. L. Yadava, Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity, J. Funct. Anal. 113 (1993), no. 2, 318–350. MR 1218099, DOI 10.1006/jfan.1993.1053
- Adimurthi and S. L. Yadava, On a conjecture of Lin-Ni for a semilinear Neumann problem, Trans. Amer. Math. Soc. 336 (1993), no. 2, 631–637. MR 1156299, DOI 10.1090/S0002-9947-1993-1156299-0
- Adimurthi and S. L. Yadava, Existence and nonexistence of positive radial solutions of Neumann problems with critical Sobolev exponents, Arch. Rational Mech. Anal. 115 (1991), no. 3, 275–296. MR 1106295, DOI 10.1007/BF00380771
- Adimurthi and S. L. Yadava, Nonexistence of positive radial solutions of a quasilinear Neumann problem with a critical Sobolev exponent, Arch. Rational Mech. Anal. 139 (1997), no. 3, 239–253. MR 1480241, DOI 10.1007/s002050050052
- Peter W. Bates, E. Norman Dancer, and Junping Shi, Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimension and instability, Adv. Differential Equations 4 (1999), no. 1, 1–69. MR 1667283
- Peter W. Bates and Giorgio Fusco, Equilibria with many nuclei for the Cahn-Hilliard equation, J. Differential Equations 160 (2000), no. 2, 283–356. MR 1737000, DOI 10.1006/jdeq.1999.3660
- Haïm Brézis and Louis Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477. MR 709644, DOI 10.1002/cpa.3160360405
- Simon Brendle, Blow-up phenomena for the Yamabe equation, J. Amer. Math. Soc. 21 (2008), no. 4, 951–979. MR 2425176, DOI 10.1090/S0894-0347-07-00575-9
- C. Budd, M. C. Knaap, and L. A. Peletier, Asymptotic behavior of solutions of elliptic equations with critical exponents and Neumann boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 117 (1991), no. 3-4, 225–250. MR 1103293, DOI 10.1017/S0308210500024707
- Luis A. Caffarelli, Basilis Gidas, and Joel Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), no. 3, 271–297. MR 982351, DOI 10.1002/cpa.3160420304
- G. Cerami, S. Solimini, and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Anal. 69 (1986), no. 3, 289–306. MR 867663, DOI 10.1016/0022-1236(86)90094-7
- G. Cerami and Juncheng Wei, Multiplicity of multiple interior peak solutions for some singularly perturbed Neumann problems, Internat. Math. Res. Notices 12 (1998), 601–626. MR 1635869, DOI 10.1155/S1073792898000385
- E. N. Dancer and Shusen Yan, Multipeak solutions for a singularly perturbed Neumann problem, Pacific J. Math. 189 (1999), no. 2, 241–262. MR 1696122, DOI 10.2140/pjm.1999.189.241
- E. N. Dancer and Shusen Yan, Interior and boundary peak solutions for a mixed boundary value problem, Indiana Univ. Math. J. 48 (1999), no. 4, 1177–1212. MR 1757072, DOI 10.1512/iumj.1999.48.1827
- Manuel del Pino, Patricio L. Felmer, and Juncheng Wei, On the role of mean curvature in some singularly perturbed Neumann problems, SIAM J. Math. Anal. 31 (1999), no. 1, 63–79. MR 1742305, DOI 10.1137/S0036141098332834
- Olivier Druet, Compactness for Yamabe metrics in low dimensions, Int. Math. Res. Not. 23 (2004), 1143–1191. MR 2041549, DOI 10.1155/S1073792804133278
- Druet,O., Robert,F. and Wei,J., On Lin-Ni’s conjecture:$N \geq 7$, preprint.
- Pierpaolo Esposito, Estimations à l’intérieur pour un problème elliptique semi-linéaire avec non-linéarité critique, Ann. Inst. H. Poincaré C Anal. Non Linéaire 24 (2007), no. 4, 629–644 (English, with English and French summaries). MR 2334996, DOI 10.1016/j.anihpc.2006.04.004
- B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), no. 8, 883–901. MR 619749, DOI 10.1080/03605308108820196
- Changfeng Gui and Nassif Ghoussoub, Multi-peak solutions for a semilinear Neumann problem involving the critical Sobolev exponent, Math. Z. 229 (1998), no. 3, 443–474. MR 1658569, DOI 10.1007/PL00004663
- Nassif Ghoussoub, Changfeng Gui, and Meijun Zhu, On a singularly perturbed Neumann problem with the critical exponent, Comm. Partial Differential Equations 26 (2001), no. 11-12, 1929–1946. MR 1876408, DOI 10.1081/PDE-100107812
- Gierer,A. and Meinhardt,H., A theory of biological pattern formation, Kybernetik (Berlin) 12 (1972), 30-39.
- Massimo Grossi and Angela Pistoia, On the effect of critical points of distance function in superlinear elliptic problems, Adv. Differential Equations 5 (2000), no. 10-12, 1397–1420. MR 1785679
- Massimo Grossi, Angela Pistoia, and Juncheng Wei, Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory, Calc. Var. Partial Differential Equations 11 (2000), no. 2, 143–175. MR 1782991, DOI 10.1007/PL00009907
- Changfeng Gui, Multipeak solutions for a semilinear Neumann problem, Duke Math. J. 84 (1996), no. 3, 739–769. MR 1408543, DOI 10.1215/S0012-7094-96-08423-9
- Changfeng Gui and Chang-Shou Lin, Estimates for boundary-bubbling solutions to an elliptic Neumann problem, J. Reine Angew. Math. 546 (2002), 201–235. MR 1900999, DOI 10.1515/crll.2002.044
- Changfeng Gui and Juncheng Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differential Equations 158 (1999), no. 1, 1–27. MR 1721719, DOI 10.1016/S0022-0396(99)80016-3
- Changfeng Gui and Juncheng Wei, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Canad. J. Math. 52 (2000), no. 3, 522–538. MR 1758231, DOI 10.4153/CJM-2000-024-x
- Changfeng Gui, Juncheng Wei, and Matthias Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincaré C Anal. Non Linéaire 17 (2000), no. 1, 47–82 (English, with English and French summaries). MR 1743431, DOI 10.1016/S0294-1449(99)00104-3
- Saïma Khenissy and Olivier Rey, A criterion for existence of solutions to the supercritical Bahri-Coron’s problem, Houston J. Math. 30 (2004), no. 2, 587–613. MR 2084920
- M. A. Khuri, F. C. Marques, and R. M. Schoen, A compactness theorem for the Yamabe problem, J. Differential Geom. 81 (2009), no. 1, 143–196. MR 2477893, DOI 10.4310/jdg/1228400630
- Yan Yan Li, On a singularly perturbed equation with Neumann boundary condition, Comm. Partial Differential Equations 23 (1998), no. 3-4, 487–545. MR 1620632, DOI 10.1080/03605309808821354
- Yanyan Li and Meijun Zhu, Yamabe type equations on three-dimensional Riemannian manifolds, Commun. Contemp. Math. 1 (1999), no. 1, 1–50. MR 1681811, DOI 10.1142/S021919979900002X
- Yan Yan Li and Lei Zhang, Compactness of solutions to the Yamabe problem. II, Calc. Var. Partial Differential Equations 24 (2005), no. 2, 185–237. MR 2164927, DOI 10.1007/s00526-004-0320-7
- Chang Shou Lin and Wei-Ming Ni, On the diffusion coefficient of a semilinear Neumann problem, Calculus of variations and partial differential equations (Trento, 1986) Lecture Notes in Math., vol. 1340, Springer, Berlin, 1988, pp. 160–174. MR 974610, DOI 10.1007/BFb0082894
- C.-S. Lin, W.-M. Ni, and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations 72 (1988), no. 1, 1–27. MR 929196, DOI 10.1016/0022-0396(88)90147-7
- Fernando Coda Marques, A priori estimates for the Yamabe problem in the non-locally conformally flat case, J. Differential Geom. 71 (2005), no. 2, 315–346. MR 2197144
- Stanislaus Maier-Paape, Klaus Schmitt, and Zhi-Qiang Wang, On Neumann problems for semilinear elliptic equations with critical nonlinearity: existence and symmetry of multi-peaked solutions, Comm. Partial Differential Equations 22 (1997), no. 9-10, 1493–1527. MR 1469580, DOI 10.1080/03605309708821309
- Manuel del Pino, Patricio Felmer, and Monica Musso, Two-bubble solutions in the super-critical Bahri-Coron’s problem, Calc. Var. Partial Differential Equations 16 (2003), no. 2, 113–145. MR 1956850, DOI 10.1007/s005260100142
- Monica Musso and Angela Pistoia, Multispike solutions for a nonlinear elliptic problem involving the critical Sobolev exponent, Indiana Univ. Math. J. 51 (2002), no. 3, 541–579. MR 1911045, DOI 10.1512/iumj.2002.51.2199
- Wei-Ming Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc. 45 (1998), no. 1, 9–18. MR 1490535
- Wei-Ming Ni, Xing Bin Pan, and I. Takagi, Singular behavior of least-energy solutions of a semilinear Neumann problem involving critical Sobolev exponents, Duke Math. J. 67 (1992), no. 1, 1–20. MR 1174600, DOI 10.1215/S0012-7094-92-06701-9
- Wei-Ming Ni and Izumi Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 44 (1991), no. 7, 819–851. MR 1115095, DOI 10.1002/cpa.3160440705
- Wei-Ming Ni and Izumi Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993), no. 2, 247–281. MR 1219814, DOI 10.1215/S0012-7094-93-07004-4
- Olivier Rey, The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal. 89 (1990), no. 1, 1–52. MR 1040954, DOI 10.1016/0022-1236(90)90002-3
- Olivier Rey, An elliptic Neumann problem with critical nonlinearity in three-dimensional domains, Commun. Contemp. Math. 1 (1999), no. 3, 405–449. MR 1707889, DOI 10.1142/S0219199799000158
- Olivier Rey, The question of interior blow-up-points for an elliptic Neumann problem: the critical case, J. Math. Pures Appl. (9) 81 (2002), no. 7, 655–696. MR 1968337, DOI 10.1016/S0021-7824(01)01251-X
- Olivier Rey and Juncheng Wei, Blowing up solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. I. $N=3$, J. Funct. Anal. 212 (2004), no. 2, 472–499. MR 2064935, DOI 10.1016/j.jfa.2003.06.006
- Olivier Rey and Juncheng Wei, Blowing up solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. II. $N\geq 4$, Ann. Inst. H. Poincaré C Anal. Non Linéaire 22 (2005), no. 4, 459–484 (English, with English and French summaries). MR 2145724, DOI 10.1016/j.anihpc.2004.07.004
- Olivier Rey and Juncheng Wei, Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity, J. Eur. Math. Soc. (JEMS) 7 (2005), no. 4, 449–476. MR 2159223, DOI 10.4171/JEMS/35
- Liping Wang and Juncheng Wei, Solutions with interior bubble and boundary layer for an elliptic problem, Discrete Contin. Dyn. Syst. 21 (2008), no. 1, 333–351. MR 2379470, DOI 10.3934/dcds.2008.21.333
- Xu Jia Wang, Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equations 93 (1991), no. 2, 283–310. MR 1125221, DOI 10.1016/0022-0396(91)90014-Z
- Xuefeng Wang and Juncheng Wei, On the equation $\Delta u+K(x)u^{(n+2)/(n-2)\pm \epsilon ^2}=0$ in $\textbf {R}^n$, Rend. Circ. Mat. Palermo (2) 44 (1995), no. 3, 365–400. MR 1388753, DOI 10.1007/BF02844676
- Zhi Qiang Wang, The effect of the domain geometry on the number of positive solutions of Neumann problems with critical exponents, Differential Integral Equations 8 (1995), no. 6, 1533–1554. MR 1329855
- Zhi Qiang Wang, High-energy and multi-peaked solutions for a nonlinear Neumann problem with critical exponents, Proc. Roy. Soc. Edinburgh Sect. A 125 (1995), no. 5, 1003–1029. MR 1361630, DOI 10.1017/S0308210500022617
- Zhi-Qiang Wang, Construction of multi-peaked solutions for a nonlinear Neumann problem with critical exponent in symmetric domains, Nonlinear Anal. 27 (1996), no. 11, 1281–1306. MR 1408871, DOI 10.1016/0362-546X(95)00109-9
- Juncheng Wei, On the interior spike layer solutions to a singularly perturbed Neumann problem, Tohoku Math. J. (2) 50 (1998), no. 2, 159–178. MR 1622042, DOI 10.2748/tmj/1178224971
- Juncheng Wei, On the boundary spike layer solutions to a singularly perturbed Neumann problem, J. Differential Equations 134 (1997), no. 1, 104–133. MR 1429093, DOI 10.1006/jdeq.1996.3218
- Juncheng Wei and Matthias Winter, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poincaré C Anal. Non Linéaire 15 (1998), no. 4, 459–492 (English, with English and French summaries). MR 1632937, DOI 10.1016/S0294-1449(98)80031-0
- Juncheng Wei and Shusen Yan, Arbitrary many boundary peak solutions for an elliptic Neumann problem with critical growth, J. Math. Pures Appl. (9) 88 (2007), no. 4, 350–378 (English, with English and French summaries). MR 2384573, DOI 10.1016/j.matpur.2007.07.001
- Juncheng Wei and Shusen Yan, New solutions for nonlinear Schrödinger equations with critical nonlinearity, J. Differential Equations 237 (2007), no. 2, 446–472. MR 2330954, DOI 10.1016/j.jde.2007.03.001
- Wei,J. and Yan,S., Infinitely many solutions for the prescribed scalar curvature problem, J. Funct. Anal. 258 (2010), 3048-3081.
- Juncheng Wei and Xingwang Xu, Uniqueness and a priori estimates for some nonlinear elliptic Neumann equations in $\Bbb R^3$, Pacific J. Math. 221 (2005), no. 1, 159–165. MR 2194150, DOI 10.2140/pjm.2005.221.159
- Meijun Zhu, Uniqueness results through a priori estimates. I. A three-dimensional Neumann problem, J. Differential Equations 154 (1999), no. 2, 284–317. MR 1691074, DOI 10.1006/jdeq.1998.3529
Additional Information
- Liping Wang
- Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
- Address at time of publication: Department of Mathematics, East China Normal University, 500 Dong Chuan Road, Shanghai, China
- Email: lpwang@math.ecnu.edu.cn
- Juncheng Wei
- Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
- MR Author ID: 339847
- ORCID: 0000-0001-5262-477X
- Email: wei@math.cuhk.edu.hk
- Shusen Yan
- Affiliation: School of Mathematics, Statistics and Computer Science, The University of New England, Armidale, NSW 2351, Australia
- Email: syan@turing.une.edu.au
- Received by editor(s): May 23, 2008
- Published electronically: April 22, 2010
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 4581-4615
- MSC (2010): Primary 35B25, 35J60; Secondary 35B33
- DOI: https://doi.org/10.1090/S0002-9947-10-04955-X
- MathSciNet review: 2645043